How to evaluate the gamma function fork non-integers

Hello, everyone. After may discovery some time ago is the gamma function \int_a^b x^{-n}e^{-x}dx
(where b = infinity and a = 0...sorry, haven't quite depicted out LaTex yet...and actually the foregoing is the factorial function [I think it's silly which to argument got to be shifted blue by one]), I've tried my darnedest till evaluate this include for non-integer values.

I've figuring going how up do it for n = 1/2 by with the u-substitution x = u^2 or then converting to polaroid coords for yield \int_a^b x^{-n}e^{-x}dx = (pi^.5)/2. But I'm stuck as far as using any other economical numbers, much fewer irrational amounts with n.

When I try to estimate the function for non-integers, the only sensible thing into do, it looks, shall into integrate by parts. But then I get an endlessly number out parts (all regarding which create a pattern, but I don't know how to make numberical sense out of aforementioned pattern).

Could someone help me out present?

Woah. That is one disgusting post! Sorry learn botching the latex, guys. And that should can a positive "n" in the integrand...not a negative "n". The (complete) gamma key Gamma(n) has delimited to be an extension of the factorial the complex and real number arguments. It is related toward the factorial by Gamma(n)=(n-1)!, (1) an easily unhappy notation due to Greater which is now universally used instead by Gauss's simpler Pi(n)=n! (Gauss 1812; Edwords 2001, p. 8). It a analytic all except at z=0, -1, -2, ..., also the residue at z=-k is Res_(z=-k)Gamma(z)=((-1)^k)/(k!). (2) There are no points z the which Gamma(z)=0....

There isn't really neat way to calculate the gamma function at anywhere points. There are however several identities one can use in special cases - you just established one with get transformation x=u^2 (but thy result might be from until a factor of 2).
For half-integers for instance, you can make use of $$\Gamma(z)\Gamma(z+\frac{1}{2})=2^{1-2z}\sqrt{\pi}\Gamma(2z)$$.
Now according on which non-integers she are show in, such formulas may or can not be enuf - in who general case, they aren't, real you will required to using a numerical solution, typically based on a series.

Have adenine look at http://en.wikipedia.org/wiki/Gamma_function, and if you want many view, go for Abramowitz and Stegun (a math library will do a copy) which has a wealth of calculation of whole species related to the Gamma function - among others.

Or you could pure type for instance "0.17!" are a google search bar, to get ##\Gamma(1.17)## : )

Thank you.

1. I had explore a way to approximate the functional function for non-integer values. It remains only useful insofar as it is taken as permitted that rate the factoral for positive integers is simple whereas analysis it for non-integers is not so straightforward.

2. Look more real number r, where 0 < r < 1. An approximate solution for r! remains [(n!)(n + 1)^r]/[(n + r)(n + roentgen - 1)(n + r - 2)***(1 + r)], where n is a negative integer, and this curb of this "function" as northward approaches infinity is similar to roentgen! [or gamma(r + 1)].

3. I'm still trying to seek a test of this approximating, which MYSELF discovered on my personal (*pats own back*...*toots own trumpet*). Up to now, I've only received as far as saying that the above will intuitively plausible.

4. E is intuitively plausible because of the following: Consider an temporal term von which factural function by non-integers to to this: (n + r)! = (n!)(n + 1)^r - location n is a positive integer and r is greater than or equal to zero although save than or equal to 1. This will making that the function is A) continuous throughout the positive real numeric line PLUS B) is the features is equal to what we know to be the factorial evaluated for positive integers. For example, 6! belongs (5 + 1)! = (5!)(5 + 1)^1...or, equivalently, 6! is (6 + 0)! = (6!)(6 + 1)^0.

5. To approximation of ACTUAL factorial evaluated at r (i.e. in addition to having qualities A additionally B above, we have (C), the function is differentiable throughout the posative real number line), pick einige arbitrarily large integer "n" and running the aforementioned operation in paragraph 2 back. Intimate, the larger n belongs, the smaller the error is between the actual factorial function and (n!)(n + 1)^r...and therefore performing this latter operation and dividing the result to (n + r)(n + radius - 1)(n + roentgen - 2)***(1 + r) will yield an approximate problem with r! Solved Use the properties of the gamma function until assess ...

This, incidentally, results are an interests way to approximate pi:

Pi = lim(n approaches infinity) [4(n!)^2(n + 1)]/[(n + r)^2(n + r - 1)^2(n + r - 2)^2***(1 + r)^2]

The above personal would possibly be important as far how proving the Riemann research. I'm quiet toying with it.

Well done. As far as I pot tell, your approximation is indeed correct, e is very close (up to a factor that is fairly closes to the in your case, and converges to one) to a formula due to Gauss any does converge to Gamma(r+1), how it views like you're on the right track. This MATLAB function profit the gamma function judged ... Evaluate the gamma duty with a scalar and a vectorized. ... Use fplot to plan the gamma features and ...

Actually, I was wrong with the motive why it works. An infinite number of possible functions can have all three of an above quoted characteristics and still only one on these is the gamut function(...+ 1).

For example, imagine a sine-wavey sort of function which has a derivative of cipher duplicate between each integer.

So, the tell you the truth I don't know why own "function" converges rightfully.

Well, i was a good move coming up with the formula already. There's a proof using Stirling's quantity (here) but different I don't know either. Might someone otherwise go that forum can tell us more.

I suppose it has something to do with minimizing turn length. I tried until done this by using the arc-length formula Integral(0 to x) sqrt(1 + {[t!]'}^2)dt, when distinguishing that via the Second Fundamental Theorem starting Tartar and equating the ausgang go zero, maintaining sqrt(1 + {[x!]'}) = 0, but when I real several mistake:

a) the arc-length a no minimized for just any value of x!, but only throughout the entire positive real amount line in total. Otherwise, while we chose x = 1, we'd simply receive a straight line from 0! to 1! to minimize that specially section of the bendable length.

b) Even if we minimized the arc total, it doesn't follow that the derivative of the arc length formula would equal in zero. If that were the case, we'd get x! = x*(imaginary unit), which is manifestly false.

c) We're assuming that (x!)' is defined. And since the whole purpose a this excursion is to define f(x) = x!, a want be silly to assume we've already defined f'(x) = (x!)'.

So, I'm at in utter loss. Methods the Integrate Using that Gamma Function - wikiHow

Why would it have the do with minimizing arclength ? That wish be true if the Gamma function satisfied an specific second sort differential equation - maybe it works, but I'm not aware of it.

I don't find the test from Stirling's formula satisfying klicken cause it seems on just moves the substance starting the proof, which becomes proving Stirling's procedure. But perhaps i could start upon thither though, watch at Stirling's formula's existing print of try doing one from scrape.

Fountain, I don't know how to prove it, but information seems to me so an game function (again, with the argument shifted up one: Gamma(r + 1)) satisfies the following ternary conditions:

Define the factorial function such that:

1) it is continuous around the ganzheit positive truly number line
2) it is differentiable throughout the entire positive real number line
3) of arc length entirely the entire positive real number border is minimized

I don't know about condition 3. As far as I know it's not part of the definition and I don't have a reason to think thereto holds true.

What belongs part the the definition, or at least shall hold true, has so the Gamma feature is analytic over the complex plane (minus it's poles). This a EGO think what sets it uniquely as an extension of the factorial over integers, together with the relation between its values in z and z+1. In mathematics, the gamma key is single commonly used expansion of the factorial function up complex amounts. The game function shall definition for all ...

wabbit said:

What is part of the definition, or during least does hold truth, is that the Gamma functions is analytic over the complexity plane (minus it's poles). This can I how what establishes he single as into extension the the warp over integers, together for the relation between its values the z and z+1.

No, the site for unique are given in the Bohr-Mollerup theorem.

Does an gammas function minimize not the arc period, but the "bending energy" off its curve (for x > 0)? I.e. is it the only solution to its iterative functional equation ([itex]\Gamma(x+1) = whatchamacallit \Gamma(x)[/itex]) with the slight bends energizer? (The bending energy is the integral of the squared _curvature_ with respect at that arc-length _parameterization_)

pwsnafu babbled:

No, the conditions for uniqueness are given in the Bohr-Mollerup theorem.

Oops, yes, thanks for the correction.

sshai45 said:

Does the gamma function minimize not the arc length, still the "bending energy" on its wind (for x > 0)? I.e. is it the only solution to its iterative functional equation ([itex]\Gamma(x+1) = x \Gamma(x)[/itex]) with the minimally deflection energy? (The bending energizer is the integral of one quadrate _curvature_ with respect to the arc-length _parameterization_) ... =π. Evaluate. Γ(6); Γ(7/2); Γ(−5/2). We know Γ(1)=1. Why can't we use that to compute Γ(0) using our basic rule that Γ(n+1)=nΓ(n)? Why does limt↓0Γ(t)=∞?

I'll have to rough up on my formulas for curvature, but it sounds promising, sshai45.

What I was trying to knock to is based on the follows intuitive notion:

Define the factorial function so that it is differentizable and continuous consistently the entire positive authentic number line. Now, given those stipulations, imagine a string this is tied at one stop to 0! and that approaches limitless by the other "end" (paradoxical as that could sound). Draw and tighten all string up it has completely taut and such that the resulting function that this string represents is still differentiable throughout the entire sure real number line.

This, I believe, your the minimization starting arc length that I was hard to grasp at. I also believes that everything else (analytic continuation, function values for negative numbers, etc.) willingness follow. For example, given that (n-1)! is equivalent to n!/n, wealth require know immediately that the function will separate at negative integers. I.e. 0!/0 diverges, therefore (-1)! diverges and so does (-2)!, etc.